We measure the angular bispectrum of the cosmic microwave background (CMB) radiation anisotropy from the COBE Differential Microwave Radiometer (DMR) 4 yr sky maps. The angular bispectrum is the harmonic transform of the three-point correlation function, analogous to the angular power spectrum, the harmonic transform of the two-point correlation function. First, we study statistical properties of the bispectrum and the normalized bispectrum. We find the latter more useful for statistical analysis; the distribution of the normalized bispectrum is very much Gaussian, while the bare bispectrum distribution is highly non-Gaussian. Then, we measure 466 modes of the normalized bispectrum, all independent combinations of three-point configurations up to a maximum multipole of 20, the mode corresponding to the DMR beam size. By measuring 10 times as many modes as the sum of previous work, we test the Gaussianity of the DMR maps. We compare the data with the simulated Gaussian realizations, finding no significant detection of the normalized bispectrum on a mode-by-mode basis. We also find that the previously reported detection of the normalized bispectrum is consistent with a statistical fluctuation. By fitting a theoretical prediction to the data for the primordial CMB bispectrum, which is motivated by slow- roll inflation, we put a weak constraint on the parameter characterizing nonlinearity in inflation. Simultaneously fitting the foreground bispectra estimated from interstellar dust and synchrotron template maps shows that neither dust nor synchrotron emission significantly contributes to the bispectrum at high Galactic latitude. We conclude that the DMR map is consistent with Gaussianity.